Definition of the derivative of a function

The definition of the derivative of a function is:

Note: Dy/Dx= f '(x)


Ex: f(x)= x^2
f '(x)= lim [(x+h)^2 - x^2]/h
h-->0
f '(x)= lim (x^2 + 2xh + h^2 - x^2)/h
h-->0
(The x^2 and the -x^2 cancel out)
f '(x)= lim h(2x+h)/h
h-->0
f '(x)= lim 2x+h
h-->0
(Plug in 0 for h since that is what the limit is approaching)
f '(x)= 2x